Quantum mechanics without quantum logic

a r X i v :q u a n t -p h /0404045v 1 7 A p r 2004Quantum mechanics without quantum logic

D.A.Slavnov

Department of Physics,Moscow State University,

Moscow 119899,Russia.E-mail:slavnov@goa.bog.msu.ru

We describe a scheme of quantum mechanics in which the Hilbert space and lin-ear operators are only secondary structures of the theory.As primary structures we consider observables,elements of noncommutative algebra,and the physical states,the nonlinear functionals on this algebra,which associate with results of single mea-surement.We show that in such scheme the mathematical apparatus of the standard quantum mechanics does not contradict a hypothesis on existence of an objective local reality,a principle of a causality and Kolmogorovian probability theory.1Introduction Quantum ?eld theory has achieved signi?cant successes in the last decades.These successes are related to creation of non-Abelian (noncommutative)gauge models.The newest quark physics has arisen on their basis.Abelian gauge model,quantum electrodynamics,was known for a long time.Transition to non-Abelian models became a qualitative leap in development of quantum ?eld theory.At the same time,this transition has not caused any essential revision of the basic concepts of quantum ?eld theory.Especially,it has not demanded any changes in logic and mathematics.In present paper the idea is carried out that transition from classical to quantum physics

is similar to transition from Abelian to non-Abelian gauge models.Of course,the quantum physics is qualitatively new theory.However,for successful development of the quantum theory it is completely not necessary to refuse the main concepts of the classical theory:the formal logic,the classical probability theory,the principle of causality,idea on the objective physical reality.

The base notions of the modern standard quantum mechanics are the Hilbert space and linear operators in this space.Von Neumann [1]mathematically precisely has formulated quantum mechanics on the basis these concepts.The matrix mechanics of Heisenberg and the wave mechanics of Schr¨o dinger are concrete realization of the von Neumann’s abstract method.

The formalism of the Hilbert space became the mathematical basis of those tremendous successes which were achieved by quantum mechanics.However,these successes have also a reverse side.There is some worship of the Hilbert space.Physicists have ceased to pay attention that the Hilbert space rather speci?c mathematical object.It has appeared a

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excellent basis for calculation of expectation values of observable quantities and their prob-abilistic distributions.At the same time,it is completely not self-evident that observables are operators in the Hilbert space.

Attempts to use the formalism of the Hilbert space for the description of the single physical phenomena not so are successful.Reasonings which are used in this case,frequently appear not indisputable.

So von Neumann[1]resorts to rather doubtful idea on”internal I”to coordinate the concept of the Hilbert space to the results of single measurements.Abandonment of the causality principle also is hard to perceive.

The same concerns the idea on the determinative in?uence of the observer onto quantum-mechanical processes.In this respect quantum mechanics is a unique subdiscipline of physics and of the science in general.The idea(see for example,[2])is put forward on the active role of conscious in the quantum phenomena.

In attempt to give slightly more objective form to similar notion Everett[3]has put forward rather exotic idea on existence in the nature of set of the parallel worlds.Happy-go-lucky the observer appears at each measurement in any one of these worlds.This idea has got very many supporters despite all extravagance.

Probably,this speci?es that though the Hilbert space rather useful mathematical object, its base role is completely not indisputable.It is not a new idea that the Hilbert space and linear operators are not primary elements of the quantum 32c90a7301f69e31433294d4ly this idea became a basis of the algebraic approach to quantum?eld theory(see for example,[4,5,6]).

2Observables and states

Base notion of classical physics is”observable”.This notion seems self-evident and does not demand de?nition.It is possible to multiply them by real numbers,to sum up and multiply together.In other words,they form a real algebra A cl.

The elements?A of this algebra are the latent form of observable variables.The explicit form of an observable should be some number.It means that the explicit form of an observ-able corresponds to the value of some functional?(?A)=A(A is a real number),de?ned on the algebra A cl.

Physically,the latent form of the observable?A becomes explicit as a result of measure-ment.This means that the functional?(?A)describes a measurement result of the observable ?A.

Experiment shows that the sum and the product of measurement results correspond to the sum and the product of observables:

?A

+?A2→A1+A2,?A1?A2→A1A2.(1)

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In this connection further there will be useful a following de?nition[7].

Let B be a real commutative algebra and??be a linear functional on B.If

??(?B1?B2)=??(?B1)??(?B2)for all?B1∈B and?B2∈B,

then the functional??is called a real homomorphism on algebra B.

Now we can introduce the second base notion of classical physics—a state of the object under consideration.

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The state is a real homomorphism on the algebra of observables.The result of any measurement of the classical object is determined by its state.

In principle,in classical physics it is possible to measure observables in any combinations. Always it is possible to pick up such system of measuring devices that measurement results of several observables will not depend on sequence in what observables are measured.For example,if we carry out measurement of an observable?A,then an observable?B,then again an observable?A and an observable?B results of repeated measurements of observables coincide with results of primary measurements.We name the corresponding measuring instruments compatible devices.

Let us pass to discussion of the situation in quantum physics.In quantum physics it seems also natural to accept”observable”as base notion.Quantum observables also possess algebraic properties.

However quantum measurements signi?cantly di?er from classical one.There are systems of compatible measuring devices not for any observables.Accordingly,in quantum physics observables are subpided into compatible(simultaneously measurable)and incompatible (additional).There are systems of compatible measuring devices for compatible observables. Such systems of devices do not exist for incompatible observables.As it is told in paper by Zeilinger[8]:”Quantum complimentarity then is simply expression of the fact that in order to measure quantities,we would have to use apparatuses which mutually exclude each other”.

For incompatible quantum observables the measurement results depend on sequence of measurement of these observables.This fact leaves traces on rules of multiplication of ob-servables.Two ways are applied.The?rst way is used in so-called Jordan algebra[9,4].In this case,observables form real commutative algebra,but the operation of multiplication is not 32c90a7301f69e31433294d4e of Jordan algebra has not led to to appreciable successes in the quantum theory.

The standard quantum mechanics is based on the other method of multiplication of observables.In this case,operation of multiplication is associative,but noncommutative. Besides product of two observables not necessarily is an observable,i.e.observables do not form algebra.

In order to use the advanced mathematical apparatus of algebras full-scale,it is conve-nient to leave for framework of the directly observable variables and to consider its complex combinations.Hereinafter,we call these combinations the dynamic variables.

Having in view of told above,we accept

Postulate1:

Dynamic variables correspond to elements of an involutive,associative,and(in general) noncommutative algebra A,satisfying the following conditions:for each element?R∈A, there exists a Hermitian element?A(?A?=?A)such that?R??R=?A2,and if?R??R=0,then ?R=0.

We assume that the algebra has a unit element?I and that Hermitian elements of algebra A correspond to observable variables.We let A+denote the set of these elements.

In the standard quantum mechanics this postulate is accepted in considerably stronger form.It is supposed that dynamic variables?R are linear operators in the Hilbert space.

Postulate2directly follows from quantum measurements

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Postulate2:

Mutually commuting elements of the set A+correspond to compatible(simultaneously measurable)observables.

Because of Postulate2,commutative subalgebras of the algebra A have an important role in the further analysis.(see[10]).

For the further it is useful to recollect de?nition of the spectrumσ(?A;A)of an element?A in the algebras A.The numberλis a point of the spectrum of the element?A if and only if the element?A?λ?I does not have an inverse element in the algebra A.Generally,an element may have di?erent spectra in an algebra and its subalgebra.However,if the subalgebra Q is maximal,σ(?Q;Q)=σ(?Q;A)for any?Q∈Q(see for example,[7]).

The Hermitian elements of the algebra A are the latent form of the observable variables. The same as in a classical case the latent form of an observable?A becomes explicit as a result measurements.Only mutually commuting observables can be are measured in inpidual experiment.Experiment shows that the same relations(1)are carried out for such observables like for classical observables.

Generalizing de?nition of a state in classical physics,we accept the central postulate of the proposed approach(see[11]).

Postulate3:

The result of the observation which we carry out on the quantum system is determined by a physical state of this system.The physical state is described by a functional?(?A)(generally, multivalued),with?A∈A+,whose restriction?ξ(?A)to each subalgebra Qξis single-valued and is a real homomorphism(?ξ(?A)=A is a real number).

The functionals?ξ(?A)can be shown to have the following properties[7]:

1)?ξ(0)=0;(2)

2)?ξ(?I)=1;

3)?ξ(?A2)≥0;

4)ifλ=?ξ(?A),thenλ∈σ(?A;Qξ);

5)ifλ∈σ(?A;Qξ),thenλ=?ξ(?A)for some?ξ(?A).

The corresponding properties of inpidual measurements are postulated in the standard quantum mechanics but are a consequence of the third postulate here.

On the other hand,properties(2.4)and(2.5)allow to construct all functionals?,ap-pearing in the Postulate3.Clearly that for construction of the functional?su?ciently to construct all its restriction?ξon subalgebras Qξ.In its turn,each functional?ξcan be constructed as follows.In each subalgebra Qξit is necessary to choose arbitrarily system G(Qξ)independent generators.Further we require?ξto be a certain mapping G(Qξ)to a real number set(allowable points of the spectra for the corresponding elements of the set G(Qξ)).On the other elements of Qξ,the functional?ξis constructed by linearity and mul-tiplicativity.Sorting out all possible mappings of the set G(Qξ)into points of the spectrum, we construct all functionals?ξ.

On other subalgebra Qξ′the functional?ξ′is constructed similarly.It is clearly that this procedure is always possible if functionals?ξand?ξ′are constructed independently.

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Thus,the set of physical states(functionals?)is completely de?ned by the algebra A(set

of its maximal real commutative subalgebras and spectra of these subalgebras).It is clearly that these functionals,generally,are multivalued.Moreover,it is possible to show[12]that

there are algebras having physical sense for which it is impossible to construct the single-valued functional

However,it is always possible to construct a functional?that is single-valued on all

observables belonging any preset subalgebra Qξ.For this,su?ce it to assign number one (setξ=1)to the subalgebra Qξand de?ne the restriction?1of?to Q1as follows.Let G(Q1)be a set of generators of Q1.We de?ne the restriction?1to be some mapping of G(Q1)

to a real number set S1.We next choose another subalgebra Q2.With Q1∩Q2≡Q12=?, we?rst construct a set of generators G12of Q12,and then supplement it with the set G21

to the complete set of generators of Q2.If?A∈G12,then?2(?A)=?1(?A).If?A∈G21,then the functional?2is de?ned such that it is a mapping of G21to some allowable set of points in the spectra of the corresponding elements of the algebra Q2.We must next exhaust all subalgebras Q i(of type Qξ)that have nonempty intersections with Q1.To construct the restriction?i of?to each Q i,su?ce it to use the recipe used for?2.By construction,such a functional?is single-valued on all elements belonging to Q1.Di?erent subalgebras Q i can have common elements that do not belong to Q1.On these elements,the functional?can be multivalued.

Physically the multivaluedness of the functional?can be justi?ed as follows.The result of observation may depend not only on an observable quantum object,but also on properties of the measuring device used for observation.A typical measuring device consists of an analyzer and a detector.The analyzer is a device with one input and several output channels.As an example,we consider the device measuring an observable?A.For simplicity,we assume that the spectrum of this observable is discrete.Each output channel of the analyzer must then correspond to a certain point of the spectrum.The detector registers the output channel through which the quantum object leaves the analyzer.The corresponding point of the spectrum is taken to be the value of the observable?A registered by the measuring device.

In general,the value not of one observable?A i but of an entire set of compatible observ-ables can be registered in one experiment.All these observables must belong to a single subalgebra Qξ.Naturally the analyzer must be constructed appropriately.The main techni-cal requirement to this construction consists in the following.Sorting of researched quantum objects on values of one of the observables belonging Qξ,should not deform sorting on values of other observable belonging Qξ.

Therefore,measuring devices can be are marked by an indexξ.We say that the corre-

sponding device belongs to the type Qξ.

Let now we want to measure value of the observable?A∈Qξ∩Qξ′.For this purpose we can use the device of the type Qξor the device of the type Qξ′.These devices have a di?erent construction.Therefore,it is absolutely not necessary that we obtain the same result if we use a di?erent devices for investigation of the same quantum object.It just corresponds to what the functional describing the physical state of the quantum object,is multivalued on the observable?A.

We cannot use both devices in one experiment as subalgebras Qξand Qξ′contain in-compatible observables.Therefore,in each concrete experiment the measurement result is single-valued.However,this result can depend not only on internal characteristics of the observable object(its physical state),but also on the characteristic(type)of the measuring device.Accordingly,the functional?describes not the value of each observable in a given

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physical state but the response of the measuring device of de?ned type,to this physical state.Here there is an essential di?erence between the proposed approach and so-called PIV model described in the review [13].In this model it is supposed that the value of each observable is uniquely predetermined for the quantum object.If the functional ?is single-valued at a point ?A ,we say that the corresponding physical state ?is stable on the observable ?A

.In this case,we can say that the observable ?A has a de?nite value in the physical state ?and this value is the physical reality.

The commutative algebra A has only one maximal real commutative subalgebra.There-fore,in this case all measuring devices belong to one type and all physical states are stable on all observables.

It is natural to impose a condition of ”minimality”on algebra A with the following physical sense.If we cannot distinguish two observables by any experiment it is one element of algebra.Therefore,we accept

Postulate 4:If sup ?ξ|?ξ(?A

1??A 2)|=0,then ?A 1=?A 2.We note that the standard quantum mechanics implies the stronger assumption that the observables coincide if all their average values coincide.

It is obvious that in case of commutative algebra,the set of the physical states passes into usual classical phase space,and a separate functional ?comes to a point in this space.However,for noncommutative algebras the physical state di?ers that is understood as a state in quantum physics.Further we use the term ”a quantum state”for latter.

Let us note that the physical state can not be ?xed uniquely in quantum measurement.Really,the devices for measurement of incompatible observables are incompatible.Therefore,we can measure the observables belonging to one maximal commutative subalgebra Q ξin one experiment.As a result we establish only values of the functional ?ξwhich is a restriction of the functional ?(the physical state).In other respects the functional ?is uncertain.Repeated measurement with use of di?erent device does not help the situation.Because the quantum object passes into a new physical state for which the values of the functional found in the ?rst experiment ?ξis useless.

We say that the functionals ?are ?ξ-equivalent if they have the same restriction ?ξon the subalgebra Q ξ.Thus,in quantum measurement we can ascertain only the class of equivalence {?}?ξto which the physical state belongs.Because the functional ?has continual set of restrictions on various maximal commutative subalgebras the set {?}?ξhas a potency of the continuum.Only this class of equivalence can claim a name ”quantum state”.

Actually there is one more restriction.Experiment shows that if the quantum object is in the quantum state corresponding {?}?ξthe measurement result of the observable ?A ∈Q ξdoes not depend on the type of used measuring device.It means that the physical state should be stable on the subalgebra Q ξ.We call such class of equivalence the quantum state and denote by Ψ(·|?ξ).

Strictly speaking,the above de?nition of the quantum state is valid only for a physical system that does not contain identical particles.Describing identical particles requires some generalization of the de?nition of the quantum state.

The measuring instrument cannot distinguish which of the identical particles entered it.Therefore,we slightly generalize the de?nition of a quantum state.Let the physical system contain identical particles.Let {?}?ξbe the set of ?ξ-equivalent functionals.We say

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that a functional?′is weakly?ξ-equivalent to the functionals?if its restriction?′ξ′on the subalgebra Qξ′coincides with the restriction?ξof the functional?on the subalgebra Qξ.

The set{?Q′}must be an image of the set{?Q}under a mapping such that the observables corresponding to one of the identical particles are changed to the respective observables

corresponding to the other particle.

Hence,the de?nition of the quantum stateΨ(·|?ξ)should refer to the set of all weakly

?ξ-equivalent functionals.Hereinafter,we let the symbol{?}?

ξdenote the set of a weakly

?ξ-equivalent functionals.

3Probability theory and quantum ensemble

Now the Kolmogorovian probability theory[14]is the most consistent and mathematically rigorous.It is usually considered that it does not approach for the description of quantum systems.In present paper the opposite opinion is protected:the Kolmogorovian probability theory very well approaches for the description of the quantum phenomena,it is necessary only to take into account peculiarity of quantum measurements[15].

We recall the foundations of Kolmogorov’s theory probability(see,for example[14,16]). The probability theory scheme is based on the so-called probability space(?,F,P).

The?rst component?is set(space)of the elementary eventsω.The physical sense of the elementary events specially is not stipulated,but it is considered that they are mutually exclusive.One and only one elementary event is realized in each test.

Besides the elementary event the concept of”event”(or”random event”)is introduced. Each event F is identi?ed with some subset of set?.It is supposed that we can ascertain, the event is carried out or failed in a experiment under consideration.Such assumption is not done about the elementary event.

Collections of subsets of the set?(including the set?and the empty set?)are supplied with the structure of Boolean algebras.Algebraic operations are:intersection of subsets, joining of them,and complement with respect to?.A Boolean algebra,closed in respect of denumerable number of operations of joining and intersection,is called aσ-algebra.

The second component of the probability space is someσ-algebra F.The set?in which the particularσ-algebra F is chosen,refers to as measurable space.Further on the measurability will play a key role.

Finally,the third component of the probability space is the probability measure P.It represents a mapping of the algebra F onto a set of real numbers satisfying the following conditions for any countable set of nonintersecting subsets F j∈F:a)0≤P(F)≤1for all F∈F,P(?)=1;b)P( j F j)= j P(F j)Let us pay attention that the probabilistic measure is de?ned only for the events which are included in the algebra F.For the elementary events the probability,generally speaking,is not de?ned.

A real random quantity on?is de?ned as a mapping X of the set?onto the extended real straight lineˉR=[∞,+∞],

X(ω)=X∈ˉR.

The setˉR is assumed to have the measurability property.The Boolean algebra F R generated by the semi-open intervals(x i,x j],i.e.,theσ-algebra that results from applying the algebraic operations to all such intervals,can be chosen as theσ-algebra in the setˉR. Let{ω:X(ω)∈F R},where F R∈F R be the subset of elementary eventsωsuch that X(ω)∈F R.The subsets F={ω:X(ω)∈F R}form theσ-algebra F in the space?.

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We consider now the application of formulated main principles of probability theory to a problem of quantum measurements.We associate an elementary event with a physical state. Accordingly,we associate the set of physical states of a quantum object with the space?. Further,we need to make this space measurable,i.e.to choose certainσ-algebra F.Here, a peculiarity of quantum measurement,which has the name”principle of complimentarity”in standard quantum mechanics,has crucial importance.We can organize each inpidual experiment only in such a way that compatible observables are measured in it.The results of measurement can be random.That is,observables correspond to the real random quantities in probability theory.

The main goal of a typical quantum experiment is to obtain the probability distributions for one or another observable quantities.We can obtain such distribution for certain collec-tion of compatible observables if an appropriate measuring device is used.From the point of view of probability theory we choose certainσ-algebra F,choosing the certain measuring device.For example,let us use the device intended for measurement of momentum of a par-ticle.Let us suppose that we can ascertain by means of this device that the momentum of particle hits an interval(p i,p j].For de?niteness we have taken a semi-open interval though it is not necessary.Hit of momentum of the particle in this or that interval is the event for the measuring device,which we use.These events are elements of certainσ-algebra.Thus, the probability space(?,F,P)is determined not only by the explored quantum object(by collection of its physical states)but also by the measuring device which we use.

Let us assume that we carry out some typical quantum experiment.We have an en-semble of the quantum systems,belonging to a certain quantum state.For example,the particles have spin1/2and the spin projection on the x axis equals1/2.Let us investigate the distribution of two incompatible observables(for example,the spin projections on the directions forming anglesθ1andθ2with regard to the x axis).We cannot measure both observables in one experiment.Therefore,we should carry out two groups of experiments which use di?erent measuring devices.”Di?erent”is classically distinct.In our concrete case the devices should be oriented by various manners in the space.

We can describe one group of experiments with the help of a probability space(?,F1,P1), another group with the help of(?,F2,P2).Although in both cases the space of elementary events?is the same,the probability spaces are di?erent.Certainσ-algebras F1and F2are introduced in these spaces to give them the property of measurability.

Mathematically[16],aσ-algebra F12that include the algebra F1as well as algebra F2 can be formally constructed.

It is said that such an algebra is generated by the algebras F1and F2.In addition to the subsets F(1)i∈F1and F(2)j∈F2of the set?,it also contains all possible intersections and unions of the subsets F(1)i∈F1and F(2)j∈F2.

But thisσ-algebra is unacceptable physically.Indeed,the event F ij=F(1)i∩F(2)j is an event in which the values of two incompatible observables of one quantum object belong to a strictly determined domain.For a quantum system,it is impossible in principle to set up an experiment that could distinguish such an event.Therefore,the probability concept does not exist for such event.In other words,there is no probability measure corresponding to the subset F ij,and theσ-algebra F12cannot be used to construct the probability space. This illustrates the following fundamental point that should be kept in mind when applying the theory of probability to quantum systems:not all mathematically possibleσ-algebras are physically acceptable.

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The probability de?nition implies numerous tests.These tests must be performed under same conditions.This applies to the object being tested as well as to the measuring instru-ment.It is obvious that the microstates of either the object or the instrument cannot be fully controlled.Therefore,the term”the same conditions”should refer to some equivalence classes for the states of the quantum object and the measuring instrument.

For a quantum object under study,such?xation is normally realized by choosing a certain quantum state.For example,in the case of spin particles,the particles with a certain spin orientation are selected.

For the measuring instrument,we also must choose a de?nite classical characteristic to be used to?x a certain equivalence class.For example,the initial single beam of particles in the instrument should split into a few well-separated beams corresponding to di?erent values of the spin projection on some distinguished direction

Thus,what corresponds to an element of the measurable space(?,F)in an experiment is the ensemble of quantum objects(which can be in a de?nite quantum state)and a measuring instrument of a certain type that allows registering an event of a de?nite form.Each such instrument can distinguish events that correspond to some set of compatible observables. As it was already mentioned,the result of inpidual an measurement may depend not only on intrinsical properties of the measured object(the physical state),but also on the type of the measuring device.In terms of the probability theory,this can be expressed as follows for a quantum system,a random variable X can be a multivalued function of the elementary eventω.

In the classical case,all observables are compatible.Accordingly,all measuring instru-ments belong to one type;therefore,the classical random quantity X is a single-valued function ofω.We note that in the quantum case,if the quantity X is interpreted as a function on the measurable space(?,F)rather than the space?,then this function is single-valued.

All this motivates us to reconsider the interpretation of the result obtained in[12],where a no-go theorem was proved.Essentially,the theorem states that there is no intrinsic char-acteristic of a particle with spin1that unambiguously predetermines the squares of the spin projections on three mutually orthogonal directions.

The conditions of the Kochen-Specker theorem are not carried out in the approach de-scribed in present paper.Really,used in paper[12]the observables(?S2x,?S2y,?S2z)are com-patible.The observables(?S2x,?S2y′,?S2z′)are also compatible.Here,the x,y′,z′directions are orthogonal among themselves,but the y,z directions are not parallel to the y′,z′directions. The observables(?S2y,?S2z)are not compatible with the observables(?S2y′,?S2z′).The devices coordinated with the observables(?S2x,?S2y,?S2z)and the(?S2x,?S2y′,?S2z′)belong to di?erent types. Therefore,these devices not necessarily should give the same result for square of spin pro-jection on the x direction.It is impossible to carry out the experiment for check of this statement,as we cannot use simultaneously two types of measuring devices in one experi-ment.

Let us consider an ensemble of physical systems which are in the quantum stateΨ(·|?ξ). We consider the physical states?of these systems as an elementary eventsωand the quantum stateΨ(·|?ξ)(the class of equivalence{?}?

)as a space?(?ξ)of the elementary

ξ

events.The observable?A is a random variable

??A?→A≡?(?A).

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Let value of the observable be measured in experiment?A∈Qξ′and the device of the type

Qξ′be used.We denote the measurable space of the elementary events by(?(?ξ),Fξ′).It corresponds to the quantum stateΨ(·|?ξ)and to theσ-algebra Fξ′(to the measuring device

of the type Qξ′).Let Pξ′be a probabilistic measure on this space,i.e.Pξ′(F)is probability of the event F∈Fξ′).

Let us consider that the event?A is realized in experiment if the registered value of the ob-servable?A is no more?A.We denote probability of this event by Pξ′(?A)=P(?:?ξ′(?A)≤?A). Knowing probabilities Pξ′(F),we can?nd probability Pξ′(?A)with the help of corresponding summations and integrations.Distribution Pξ′(?A)is marginal for the probabilities Pξ′(F).

The observable?A can belong not only the subalgebra Qξ′but also other maximal subal-gebra Qξ”.Therefore,for de?nition of probability of event?A we can use the device of the type Qξ”.In this case for probability we could obtain other value Pξ”(?A).However,experi-ment shows that the probabilities do not depend on a used measuring device.Therefore,we should accept one more postulate.

Postulate5:

Let the observable be?A∈Qξ′∩Qξ”,then the probability to?nd out the event?A for the system which are in the quantum stateΨ(·|?ξ),does not depend on of the type used device, i.e.P(?:?ξ′(?A)≤?A)=P(?:?ξ”(?A)≤?A).

Therefore,although the functional?can be multivalued,we have the right to use nota-tions P(?:?(?A)≤?A)for probability of the event?A.

Let we have ensemble of the quantum systems which are in the quantum stateΨ(·|?ξ). For this ensemble we carry out a series of experiments in which the observable?A is measured. We deal the?nite set of the physical states in any real series.In the ideal series this set

can be denumerable.We let{?}A?

ξdenote a random denumerable sample in the space?(?ξ)

which contains all the physical states of the real series.By the law of the large numbers(see for example,[16])the probabilistic measure P?

A

in this sample is uniquely determined by the probabilities P(?:?(?A)≤?A).

The probabilistic measure P?

A determines average value of the observable?A in the sample

{?}A?ξ:

?A = {?}A?ξP?A(d?)?(?A)≡Ψ(?A|?ξ).(3)

This average value does not depend on concrete sample,and is completely determined by the quantum stateΨ(·|?ξ).

Formula(3)de?nes a functional(quantum average)on set A+.We denote this functional also byΨ(·|?ξ).The totality of all quantum experiments speci?es that we must accept the following postulate.

Postulate6:

The functionalΨ(·|?ξ)is linear on the set A+.

It implies that

Ψ(?A+?B|?ξ)=Ψ(?A|?ξ)+Ψ(?B|?ξ)also in the case where[?A,?B]=0.

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Any element?R of the algebra A is uniquely represented as?R=?A+i?B,where?A,?B∈A+. Therefore,the functionalΨ(·|?ξ)can be uniquely extended to a linear functional on A:Ψ(?R|?ξ)=Ψ(?A|?ξ)+iΨ(?B|?ξ).

Let us de?ne norm of the element?R by equality

?R 2=sup?ξ?ξ(?R??R).

Such de?nition is allowable.Due to the property(2.3)we have ?R 2≥0.It follows from property(2.1)and Postulate4that if ?R 2=0,then?R=0.Further,by virtue of de?nition of the probabilistic measure

Ψ(?R??R|?ξ)= {?}R?R?ξP?R??R(d?)?(?R??R)≤sup?ξ′?ξ′(?R??R).

If?R??R∈Qξ,thenΨ(?R??R|?ξ)=?ξ(?R??R).Therefore,

?R 2=sup?ξ?ξ(?R??R)=sup?ξΨ(?R??R|?ξ).

BecauseΨ(·|?ξ)is a linear positive functional,the Cauchy-Bunyakovskii-Schwars inequality

|Ψ(?R??S|?ξ)Ψ(?S??R|?ξ)|≤Ψ(?R??R|?ξ)Ψ(?S??S|?ξ).

is satis?ed.

Because?ξ([?R??R]2)=[?ξ(?R??R)]2,we have ?R??R = ?R 2.Therefore,if we complete the algebra A with respect to the norm · ,then A turns out C?-algebra[18].

Thus,a necessary condition of consistency of the postulate of the linearity is the following strengthening of Postulate1.

Postulate1:

The set of the dynamical variables is algebra which can be is equipped with structure C?-algebra.

The reason that we did not accepted this formulation of the?rst postulate initially because it follows from the experiment that the observables have algebraic properties and the quantum mean values have the linearity property.But mathematical relations included in the de?nition of a C?-algebra are not directly related to the experiment.

4Time evolution and the ergodicity condition

In the standard quantum mechanics the time evolution is determined by the unitary auto-morphism

?A(t)=?U?1(t)?A(0)?U(t)?A(0)=?A,(4) where?A(t)and?U(t)are operators in some Hilbert space.The operators of evolution?U(t) realize unitary representation of one-parameter group.But for(4)to preserve its physical meaning,it su?ces to consider?A(t)and?U(t)as elements of some algebra(in particular,of A).

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In our case,the evolution equation can be rewritten in terms of physical states.We therefore accept

Postulate7.

A physical state of a quantum system evolves in time as

?(?A)→?t(?A)≡?(?A(t)),(5) where?A(t)is de?ned by Eq.(4).

Equation(5)describes time evolution of a physical state entirely unambiguously.It is a di?erent story,though,that an observation allows determining the initial value?(?A)of a

functional only up to its belonging to a certain quantum state{?}?

ξ.Most of our predictions

regarding the time evolution of a quantum object are therefore probabilistic.In addition, Eqs.(4)and(5)are valid only for systems that are not exposed to?rst-class actions(in von Neumann’s terminology[1]),i.e.,do not interact with a classical measuring device.

We now return to the?fth and the sixth postulates.From the experimental standpoint, these postulates are well justi?ed.But it is not quite clear whether they can be realized within the mathematical scheme considered here.It turns out that these postulates can be related to the time evolution of the quantum system.For this,we must impose restrictions on the elements?U(t).

We now accept

Postulate8.

The elements?U(t)are unitary elements,which have an integral representation of the form

?U(t)= ?p(dE)exp[iEt],(6)

where?p(dE)are orthogonal projectors.The spectrum of?U(t)contains at least one discrete nondegenerate value E0.

Hereinafter integrations(and also limits)on algebra A are understood in sense of the weak topology of C?-algebra[18].

Somewhat conventionally,we can represent?p(dE)as

?p(dE)=?p p(dE)+?p c(dE)= n?p nδ(E?E n)dE+?p c(dE.)(7)

Here?p p(dE)and?p c(dE)concern to point and continuous spectrums,accordingly.Besides,?p n?p m=?p m?p n=0for m=n,?p n?p c(dE)=?p c(dE)?p n=0.The sum over n in(10)must necessarily involve at least one term(n=0)with a nondegenerate value E0.

In addition to this last restriction,other requirements are always assumed in considering any quantum mechanics model.Requiring a discrete point in the spectrum does not seem too restrictive either.For example,a one-particle quantum system can have a purely continuous energy spectrum.But it can be considered as a one-particle state of an extended system that can also be in the vacuum state in addition to the one-particle state.The energy spectrum of the extended system already has a discrete nondegenerate point in the spectrum.

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By the nondegeneracy of E0,we assume that the projector?p0in decomposition(10)is one-dimensional.A projector?p is said to be onedimensional if it cannot be represented as

?p= α?pα,?pα=?p,?p?pα=?pα?p=?pα.

Let us remark that,if two elements?A1and?A2of algebra A have identical spectral representation,then they obey the fourth postulate.Therefore,such elements coincide.

We call physical state?α0a ground state if?α0(?p0)=1.

Statement.If?A∈A+,then?A0≡?p0?A?p0has the form?A0=?p0Ψ0(?A),whereΨ0(?A)is the linear,positive functional.It satis?es the normalization conditionΨ0(?I)=1.

Proof.Because[?A0,?p0]=0it follows that?A0and?p0have the common spectral decjmposition of unity.Since the projector?p0is one-dimensional,the spectral decomposition ?A

has the form?A0=?p0Ψ0(?A)+?A′0where?A′0is orthogonal to?p0.Therefore,?A0=?p0?A0=?p0?p0Ψ0(?A)+?p0?A′0=?p0Ψ0(?A).

Linearity:

?p0Ψ0(?A+?B)=?p0(?A+?B)?p0=?p0Ψ0(?A)+?p0Ψ0(?B.)

From here followsΨ0(?A+?B)=Ψ0(?A)+Ψ0(?B).

By linearity,the functionalΨ0(?A)can be expanded to the algebra A,Ψ0(?A+i?B)=Ψ0(?A)+iΨ0(?B),where?A,?B∈A+.

Positivity:

Ψ0(?R??R)=?α0(?p0Ψ0(?R??R))=?α0(?p0?R??R?p0)≥0,

by virtue of the property(2.3).

Normalization:

Ψ0(?I)=?α0(?p0Ψ0(?I))=?α0(?p0?I?p0)=1

.

To?nd the physical meaning of the functionalΨ0,it is necessary to consider an element ˉA in the algebra A that corresponds to an observable?A averaged in time.

ˉA=lim

L→∞1

2L L?L dt?U?1(t)?A?U(t.)(8)

It is possible to show(see[19])that

Ψ0(?A)=?α0(ˉA).

That is,the value of the observable?A in the quantum ground stateΨ0is equal to the value of the observableˉA in the physical ground state?α0.This value is the same in all physical ground states.

The functionalΨ0has all the properties that must be possessed by a functional determin-ing quantum mean values.It is linear,is positive,and is equal to unit on the unit element. In addition it is continuous,as a linear functional on the C?-algebra.Therefore,we can accept the ergodicity axiom.

Postulate9:

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The mean value of an observable?A in the quantum ground state is equal to the value of the observableˉA(the observable?A average in time)in any physical ground state.

Thus,averaging in quantum ensemble can be reduced to averaging in time.We note that the Postulate9does Postulates5and6super?uous.

To construct the standard mathematical formalism of quantum mechanics,we can now

use the canonical construction of Gelfand-Naimark-Segal(GNS)(see,e.g.,

[4]).

We consider two elements?R,?S∈A equivalent if the conditionΨ0 ?K?(?R??S) =0 is valid for any?K∈A.We letΦ(?R)denote the equivalence class of the element?R and consider the set A(Ψ0)of all equivalence classes in A.We make A(Ψ0)a linear space setting aΦ(?R)+bΦ(?S)=Φ(a?R+b?S).The scalar product in A(Ψ0)is de?ned as Φ(?R),Φ(?S) =Ψ0(?R??S).This scalar product generates the norm Φ(?R) 2=Ψ0(?R??R)in A(Ψ0).

Completion with respect to this norm makes A(Ψ0)a Hilbert space.Each element?S of the algebra A is uniquely assigned a linear operatorΠΨ(?S)acting in this space asΠΨ(?S)Φ(?R)=Φ(?S?R).

5Examples

To illustrate the above,we consider two simple examples.

First we consider a quantum system whose observable quantities are described by Her-mitian2×2matrices.The Hamiltonian?H and the elements?p0,?A are given by

?H= E000?E0 ,?p0= 0001 ,?A= a b c d .

Obviously,?p0Ψ0(?A)=?p0?A?p0=?p0d,i.e.,

Ψ0(?A)=d.(9) In addition,

ˉA=lim

L→∞

1

4

+b b? 1/2,r0=a+d

2r ,n2=

b?b?

2r

.

The commutator of the matrices?τ(ˉn),?τ(ˉn′)is nonvanishing forˉn′=±ˉn.Therefore,each matrix?τ(ˉn)(up to a sign)is a generator of a real maximal commutative subalgebra.Because ?τ(ˉn)?τ(ˉn)=?I,the spectrum of?τ(ˉn)consists of two points±1.

14

Let{f(ˉn)}be the set of all functions taking the values±1and such that f(?ˉn)=?f(ˉn).

A physical state is described by a functional whose value coincides with one of the points in the spectrum of the corresponding algebra element.For each point of the spectrum,there exists an appropriate functional.Therefore,to the set of physical states,there corresponds a set of functionals de?ned by

?(?τ(ˉn))=f(ˉn).

Taking properties(2)into account(which must be possessed by each physical state),we obtain

?(?A)=r0+r f(ˉn).(12) The ground state is any functional?0αsuch that

f(n1=0,n2=0,n3=1)=?1.

Substituting the elementˉA((10))in(12),we obtain

?0α(ˉA)=a+d

2

=d.

This agrees with(9).

Because all maximal commutative subalgebras have one independent generator in this model,the physical states are described by the single-valued functionals.If there were several generators,then multivalued functionals would inevitably arise.It corresponds to the result received(in other terms)by Kochen and Specker[12].

As the second example we consider a harmonic oscillator.

In this case the algebra of dynamical variables is algebra with two noncommuting Her-mitian generators?Q and?P satisfying the commutative relation

[?Q,?P]=i.

Time evolution in the algebra controls the Hamiltonian

?H=1/2(?P2+ν2?Q2).

The elements?Q,?P and?H are unbounded.Therefore,they do not belong to the C?-algebra. However,their spectral projectors are elements of the C?-algebra,i.e.?Q,?P and?H is the elements joined to the C?-algebra.Thus,in this case the A-algebra is a C?-algebra with the joined elements.

It is convenient to turn from the Hermitian elements?Q and?P to elements

?a?=1

2ν

(ν?Q+i?P),?a+=

1

2ν

(ν?Q?i?P)

with the commutative relation

[?a?,?a+]=1(13) and simple time dependence

?a?(t)=?a?exp(?iνt),?a+(t)=?a+exp(+iνt).

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Let us calculate a generating functional for Green functions.In standard quantum me-chanics the n-time Green function is de?ned by the equation

G(t1,...t n)= 0|T(?Q(t1)...?Q(t n))|0 ,(14) where T is a operator of chronological ordering and|0 is a quantum ground state.

According to Postulate9in the proposed approach the Green function is de?ned by the equation

?p0T(?Q(t1)...?Q(t n))?p0=G(t1,...t n)?p0,(15) where?p0is a spectral projector?H corresponding to the minimal value of energy.

It is easy to make sure that?p0can be represented in form

?p0=lim

r→∞

exp(?r?a+?a?).(16) As well as earlier,here the limit is understood in sense of weak topology of the C?-algebra.

First we prove the auxiliary statement:

?J=lim

r1,r2→∞

exp(?r1?a+?a?)(?a+)k(?a?)l exp(?r2?a+?a?)=0.(17)

LetΨbe an any positive linear functional.Then

Ψ(?J)=lim

r1,r2→∞

exp(?r1k?r2l)Ψ((?a+)k exp(?r1?a+?a?)exp(r2?a+?a?)(?a?)l).(18)

Here,we have used a continuity of the functionalΨand the commutative relation(13). Further,

|Ψ(?J)|≤lim

r1,r2→∞

exp(?r1k?r2l)|Ψ((?a+)k exp(?r1?a+?a?)exp(?r1?a+?a?)(?a?)k)|1/2(19)×|Ψ((?a+)l exp(?r2?a+?a?)exp(?r2?a+?a?)(?a?)l)|1/2

≤lim

r1,r2→∞

exp(?r1k?r2l)|Ψ((?a+)k(?a?)k)|1/2|Ψ((?a+)l(?a?)l)|1/2

Here we considered that exp(?r?a+?a?) ≤1.It follows from(19)that|Ψ(?J)|=0,i.e.it is valid(17).

We now prove(16).In terms of the elements?a+,?a?the Hamiltonian?H has form ?H=ν(?a+?a?+1/2).According(17),

lim r1,r2→∞exp(?r1?a+?a?)?H exp(?r2?a+?a?)=

ν

2

lim

r→∞

exp(?(r)?a+?a?).

It proves the equality(16).

It follows from the equation(15)that

G(t1,...t n)?p0= 1δj(t1)...δj(t n)?p0T exp i ∞?∞dt j(t)?Q(t) ?p0 j=0.(20) By the Wick theorem(see[20])

T exp i ∞?∞dt j(t)?Q(t) =(21) =exp 1δ?Q(t1)D c(t1?t2)δ

Here::is operation of normal ordering and

D c(t1?t2)=1ν2?E2?i0

Carrying out a variation over?Q in the right-hand side(21)and taking into account(17), we have

?p0T exp i ∞?∞dt j(t)?Q(t) ?p0=exp ?1

2i ∞?∞dt1dt2j(t1)D c(t1?t2)j(t2) .

Comparing with(20),we obtain

G(t1...t n)= 1δj(t1)...δj(t n) j=0,

where

Z(j)=exp i

and consider the combination

I=|E(a,b)?E(a,b′)|+|E(a′,b)+E(a′,b′)|=(23)

= P(dλ)A a(λ)[B b(λ)?B b′(λ)] + P(dλ)A a′(λ)[B b(λ)+B b′(λ)] .

The equalities

A a(λ)=±1/2,

B b(λ)=±1/2(24) are satis?ed for any directions a and b.Therefore,

I≤ P(dλ)[|A a(λ)||B b(λ)?B b′(λ)|+|A a′(λ)||B b(λ)+B b′(λ)|]=(25)

=1/2 P(dλ)[|B b(λ)?B b′(λ)|+|B b(λ)+B b′(λ)|].

Due to the equality(24)for eachλone of the expressions

|B b(λ)?B b′(λ)|,|B b(λ)+B b′(λ)|(26) is equal to zero and the other is equal to unity.Here it is crucial that the same value of the parameterλappears in both expressions.Hence,the Bell inequality(CHSH)then follows:

I≤1/2 dP(λ)=1/2.(27) The correlation function can be easily calculated within standard quantum mechanics. We obtain

E(a,b)=?1/4cosθab,

whereθab is the angle between the directions a and b.For the directions a=0,b=π/8,

a′=π/4,b′=3π/8we have

√

I=1/

Accordingly,the equation(23)now has the form

I= {?}AB?ξP?A?B(d?)?(?A?B)? {?}AB′?ξP?A?B′(d?)?(?A?B′) +

+ {?}A′B?ξP?A′?B(d?)?(?A′?B)+ {?}A′B′?ξP?A′?B′(d?)?(?A′?B′) .

If the directions a and a′(b and b′)are not parallel to each other,then the observables ?A?B,?A?B′,?A′?B,?A′?B′are mutually incompatible.Therefore,there is no physically acceptable universalσ-algebra that corresponds to the measurement all these observables.It follows that there is no probability measure common for these observables.Besides,the sets{?}AB?

ξ

, {?}AB′?ξ,{?}A′B?ξ,{?}A′B′?ξare di?erent random denumerable samples from continual space ?(?ξ).The probability of their crossings is equal to zero.Therefore,the probability of occurrence of combinations of the type(26)is equal to zero.As a result,the reasoning which have led to to an inequality(27),appears unfair for the physical states.

Thus,the hypothesis that local objective reality does exist in the quantum case does not lead to the Bell inequalities.Therefore,the numerous experimental veri?cations of the Bell inequalities that have been undertaken in the past and at present largely lose theoretical grounds..

7The possible carrier of”the objective local reality”

In the previous sections of the paper we tried to show that,contrary to a popular belief, the mathematical formalism of the standard quantum mechanics does not contradict the hypothesis on existence of an objective local reality.In the developed approach this reality is identi?ed with concept”the physical state”.The mathematical essence of this concept is de?ned quite uniquely(the functional?).It would be desirable to have some physical?lling of this concept.

In a present section we heuristically consider one of the variants[23]of such?lling.This variant is not unique.At the same time,it gives simple and obvious interpretation to the problem phenomenon such as a collapse of the quantum state.

Within the framework of the algebraic approach the quantum object is a?nite region in the space-time,which the certain noncommutative local algebra(algebra of quantum local observables)is associated with.

On the other hand,the quantum object is a source of some?eld.Obviously,any quantum object is a source of the gravitational?eld.Till now all attempts consistently to quantize the gravitational?eld were not crowned with success.Probably,it is related to that the gravitational?eld is classical,i.e.the corresponding algebra of observables is commutative.

Probably,quantum objects are also sources of other classical?elds,in particular,the classical electromagnetic?eld.This?eld should be very weak.In this case for microobjects it is unobservable on background of the quantum electromagnetic?eld.

It is natural to assume that the classical?eld radiated by a microobject,is coherent to this object.Thus,it is a carrier of the information about the object.If the microobject is multipartial then the radiated?eld is coherent to both the separate parts of the object,and to all collective.In this case the?eld is a carrier of the information about correlations.

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Macroscopic bodies can act on the radiated classical ?eld essentially.This action can be two types.The ?rst type is action which destroys the coherence of the ?eld.Such action is irreversible.The second type is action which preserves the coherence.So the mirror acts on 32c90a7301f69e31433294d4ually,both type of interactions are present at measuring devices.The coherence is preserved in the analyzer and destroyed in the detector.

On account of weaknesses the classical ?eld exercises negligibly small e?ect on both micro-and macroobjects.Unique exception is the microobject,coherent to this ?eld.In this case the action is resonant.Therefore,even very small action can lead to appreciable result if the quantum object is in the state of the bifurcation.That is,in the state where without taking into account this action several variants of the further evolution are possible.In this case such action can play a role of random force which forces the quantum object to choose one of the variants.In this sense the classical ?eld can be considered as a pilot ?eld.It is possible to assume that the con?guration of the classical ?eld,coherent to the quantum object,is the objective reality which determines the physical state of this object.

For an illustration of this phenomenon we discuss experiment whose scheme is represented on Figure 1.

-??

?

? ? ?

"!# "!# 1

23

4D 1D 2D 3@@@@@@@@Figure 1.

The device consists of four mirrors (1,2,3,4)and three detectors (D 1,D 2,D 3).The mirrors 1and 4are semipermeable.The detectors D 1and D 3are necessary only for registration of photons.The detector D 2plays central role in the phenomenon of collapse.At the device the photon and the coherent classical ?eld either are re?ected from mirrors,or pass through them.After re?ection from the mirrors the phase changes by π/2,at passage through a semipermeable mirror the phase does not change.

Second,the mirror 1is a point of the bifurcation for the photon.Without taking into account interaction with the oscillations excited by the classical ?eld in the mirror,both channels for the photon are equally allowable.Oscillations are very weak but they are coherent to the photon.Therefore,interaction is resonant.Due to this interaction the information which is stored in the classical ?eld (the physical state)dictates to the photon a choice of the channel.

From the point of view of quantum mechanics this choice is random.The fact is that quantum mechanics deals not with physical state,but only with its generalized characteristic —quantum state.The various con?gurations of the coherent classical ?eld correspond to

20

the same quantum state.On the other hand,with the help of preliminary measurements we can receive an information only about the quantum state.Therefore,the choice of the routes by the photon is random for us.

The phases of the photon and separate parts of the classical?eld can vary when they pass channels,but their coherency is kept.According to rules of the classical optics in the mirror 4the separate parts of the classical?eld interfere so that after the mirrors4the?eld does not propagate to the detector D3.Physically the classical?eld raises the small collective oscillations in the mirror4coherent to the?eld.The scattering occurs on these oscillations. The photon also is coherent to these oscillations and scatters the same as the classical?eld. Therefore,it also does not hit the detector D3.

We now consider the second variant of the experiment when the detector D2is switched on.In the mirror1everything happen the same way as in the?rst variant.Two scenarios are further possible,in which the photon goes by the route1-3-4or does by the route1-2-4.

In the scenarios with the route1-3-4the photon hits the detector D2.There the photon participates in interaction with the classical device.The device goes out of unstable equilib-rium due to interaction with the photon.The catastrophic process develops in the device. This process has macroscopically observable result and the quantum object is registered.

The detector exerts strong action on the photon.Its state changes,and it loses a co-herency with earlier radiated classical?eld.Again radiated classical?eld is coherent to the photon in the new state.At the same time,the?eld in the channel1-2-4is not coherent to the photon.Because only the coherent classical?eld determines the physical state of the quantum object,the?eld in the channel1-2-4is e?ectively lost for the photon.

This process results in a sharp modi?cation of the coherent classical?eld of the quantum object.The quantum state of the object also changes sharply thereof.This phenomenon has all features of the collapse.However,any inconsistency with a relativity theory does not arise,as in the channel1-2-4the classical?eld does not change.The modi?cations happen in the channel1-3-4.Thus,the classical?elds in the channels1-2-4and1-3-4do not disappear in the collapse,but these?elds lose the coherence with each other.Therefore,in the mirror4 interference is 32c90a7301f69e31433294d4tter corollary agrees with the corollary adduced in the review[24].

Let us consider now the second scenario in which the photon goes by the route1-2-4,and the photon-free classical?eld goes through the detector D2.This?eld exercises negligibly small e?ect on the detector.In this case the cause generating catastrophic process in the detector is absent.Any macroscopicaly observable of the reaction of the device is not present.

On the contrary,the action of the detector a?ects strongly the classical?eld in the channel1-3-4.This?eld loses coherence with the?eld and the photon in the channel1-2-4. The situation is the same as in the?rst scenario.

Quite similarly it is possible to interpret so-called delayed-choice experiment[25].It is usually considered that this experiment testi?es to absence of the local physical reality in the quantum phenomena.

We now can look at the experiment double-slits[26]in a new fashion.The distinct inter-ference pattern is observed in this experiment.If to reject any verbal ornaments standard interpretation of this experiment is reduced to the following.Up to the slits the inpisible quantum object passes simultaneously through the slits pided by the macroscopic distance then it again becomes inpisible.

The proposed approach allows to interpret this experiment much more evidently.The inpisible quantum object hits one of the slits and scatters on it.It can scatter in region behind the slit or in the opposite side.From the point of view of the standard quantum

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